## Regular polytopes |

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### Contents

IX | 8 |

Polyhedral kaleidoscopes | 21 |

Real fourspace and the unitary plane | 29 |

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27 lines 3-space apeirogon automorphisms belong binary Cayley diagram cells central inversion centre circle complex numbers complex polygon congruent conjugate coordinates cos2 coset Coxeter and Moser cube cubic surface cyclic group deduce defined denote derived diagonals diameters dihedral group edges Eisenstein integers elements equation Euclidean EXERCISES expressed faces finite group frieze pattern given gons graph group of order half-turn Hermitian form Hessian polyhedron honeycomb hyperplanes imply inner product instance intersecting isometry Journal of Mathematics matrix mirrors multiplied non-starry notation obtain orthogonal orthoscheme pairs parallel permutations perpendicular Petrie polygon points polyhedra presentation pure quaternion quotient group r-symmetric R1 and R2 reciprocal reflection group reflection of period regular complex polygons regular polytope relations represent rotation satisfy Schwarz triangle Shephard Similarly skew spherical triangle square subgroup of index symbol symmetry group tessellation tetrahedron Theorem three reflections unitary transformations vector vertex figure Witting polytope yields